Control of Hopf Bifurcation and Chaos in a Delayed Lotka-Volterra Predator-Prey System with Time-Delayed Feedbacks

A delayed Lotka-Volterra predator-prey system with time delayed feedback is studied by using the theory of functional differential equation and Hassard’s method. By choosing appropriate control parameter, we investigate the existence of Hopf bifurcation. An explicit algorithm is given to determine the directions and stabilities of the bifurcating periodic solutions. We find that these control laws can be applied to control Hopf bifurcation and chaotic attractor. Finally, some numerical simulations are given to illustrate the effectiveness of the results found.


Introduction
Lotka-Volterra system is one of the most classical and important systems in the field of mathematical biology. Since the word of Volterra, there have been extensively detailed investigations on Lotka-Volterra system including stability, attractivity, persistence, periodic oscillation, bifurcation and chaos (see [1][2][3][4][5][6] and the references therein). In particular, the properties of periodic solutions arising from the Hopf bifurcation are of great interest [7][8][9][10]. But the study on chaos control of Lotka-Volterra system is scarce.
Reference [3] and the references therein proposed that, for a two-species competition system with delayṡ when is big enough, the chaotic behavior may occur.
For example, Yan and Zhang [9] investigated the following delayed prey-predator system with a single delay: where 1 , 2 , 11 , 12 , 21 , and 22 are all positive constants. The delay ≥ 0 denotes the gestation period of the predator. Their results show that, taking as the bifurcation parameter, when passes through a certain critical value, the positive equilibrium loses its stability and Hopf bifurcation takes place. Furthermore, when takes a sequence of critical values containing the above critical value, the positive equilibrium of system (2) will undergo a Hopf bifurcation. With the further increase of the delay, the system will show the chaotic phenomenon (see Figure 1). In the sense of biology, chaotic behavior sometimes is to the disadvantage of virtuous cycle and develop of the ecosystem, so we want to control this chaos phenomenon and create periodic orbits. So far, many researchers have proposed chaos control schemes in recent years [11][12][13][14][15][16]. For example, Song and Wei in [17] investigated the chaos phenomena of Chen's system using the method of delayed feedback control. Their results show that, when the controlling parameter to be some value, taking the delay as the bifurcation parameter, when passes through a certain critical value, the stability of the equilibrium will be changed from unstable to stable, chaos vanishes, and a periodic solution emerges.
To the end of controlling chaos in system (2), stimulating by the works of above, we add some delayed feedback terms to system (2), that is, the following delayed feedback control system:̇( where ( = 1, 2) denote the capture coefficient when < 0 (or release coefficient when > 0). By choosing and as bifurcation parameter, we get the conditions under which Hopf bifurcation occurs. And then, we derive the explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions. At last we will give some example showing that when is fixed, with increasing, the stability of the positive equilibrium will be changed, chaos vanishes, a periodic solution occurs.
This paper is organized as follows. In Section 2, we first focus on the stability and Hopf bifurcation of the positive equilibrium. In Section 3, we derive the direction and stability of Hopf bifurcation by using normal form and central manifold theory. Finally in Section 4, numerical simulations are performed to support the stability results.

Stability and Hopf Bifurcation Analysis with Delayed Feedback Control
In this section, by analyzing the characteristic equation of the linearized system of system (3) at the positive equilibrium, we investigate the stability of the positive equilibrium and the existence of the local Hopf bifurcations occurring at the positive equilibrium. To guarantee that system (3) has always a positive equilibrium, throughout this section, we assume that the coefficients of system (3) satisfy the following condition: (H1) 21 1 − 11 2 > 0.
(H2) Equation (13) has at least one positive real root. Without loss of generality, we assume that (14) has four positive roots, defined by 1 , 2 , 3 , 4 , respectively. Then (12) has four positive roots From (11), we have If we denote where = 1, . . . , 4; = 0, 1, . . ., then ± is a pair of purely imaginary roots of (8) with = ( ) . Define Let ( ) = ( ) + ( ) be the root of (8) near = ( ) satisfying ( ( ) ) = 0, ( ( ) ) = . Substituting ( ) into (8) and taking the derivative with respect to , we have It follows that Then where Denote = 2 + 2 ; then > 0, and we have and note that In order to get the main results, it is necessary to make the following assumption. Note that when = 0, (8) becomes From [9], we know that all the roots of (26) have negative real parts; hence, the positive equilibrium * is locally asymptotically stable for = 0. Then, we can employ a result of Ruan and Wei [18] to analyze (8). For the convenience of the reader, we state it as follows.
From Lemma 1 and the above assumption, we can obtain the following theorem. ( 2) and ( 3) hold; then the following results hold true. (3) exhibits Hopf bifurcation at the positive equilibrium for = 0 .

Direction and Stability of the Hopf Bifurcation
In this section, we obtain the conditions under which a family periodic solutions bifurcate form the steady state at the critical value of . As pointed out by Hale and Verduyn Lunel [19] and Hassard et al. [20], it is interesting to determine the direction, stability, and period of these periodic solutions bifurcating from the steady state. Following the ideal of [20], we derive the explicit formulae for determining the properties of the Hopf bifurcation at the critical value of using the normal form and the center manifold theory. For the sake of simplicity of notation, we denote the critical values ( ) as , and when = , we denote the pair of purely imaginary roots of (8) as ± . Let = − ; then = 0 is the Hopf bifurcation value of system (5). In the following, we consider the equivalent system (6). Let = ; then the system (5) can be rewritten as a functional differential equation in C([−1, 0], R 2 ): In fact, we can choose where denotes Dirac-delta function. For ∈ C([−1, 0], R 2 ), define Then when = 0, the systeṁ is equivalent to the system (29), where ( ) = ( + ), ∈ and a bilinear inner product where ( ) = ( , 0), and let = (0); then and * are adjoint operators. By the discussion in Section 2, we know that ± are eigenvalues of . Thus, they are also eigenvalues of * . We first need to compute the eigenvector of and * corresponding to and − , respectively. Suppose that ( ) = (1, ) is the eigenvector of corresponding to . Then ( ) = ( ). It follows from the definition of , , and ( , ) that be the eigenvector of * corresponding to − . By the definition of * , we can compute * = 12 * /( + 2 − ( 22 * + 2 ) ). In order to assure ⟨ * ( ), ( )⟩ = 1, we need to determine the value of . From (36), we have Thus, we can choose such that ⟨ * ( ), ( )⟩ = 1, ⟨ * ( ), ( )⟩ = 0.
In the following, we first compute the coordinates to describe the center manifold 0 at = 0. Define where and are local coordinates for center manifold 0 in the direction of and . Note that is real if is real. We consider only real solutions. For the solution ∈ 0 , since = 0, we havė   ] .
(46) Substituting 1 (0), 2 (0), 1 (−1), and 2 (−1) into the above equation and comparing the coefficients with (43), we get In order to assure the value of 21 , we need to compute 20 ( ) and 11 ( ). From (34) and (40), we havė where ( , , ) = 20 ( ) Notice that near the origin on the center manifold 0 , we havė where 1 = ( (1) 1 , (2) 1 ) ∈ R 2 is a constant vector. In the same way, we can also obtain 11 ( ) = − 11 (0) In what follows, we will compute 1 and 2 . From the definition of and (51), we have From (48) and (49), we have Substituting (55) and (59) into (57) and noticing that we obtain which leads to where It follows that Similarly, substituting (56) and (60) into (58), we can get the formula of 2 , where Thus, we can determine 20 ( ) and 11 ( ). Furthermore, we can determine each . Therefore, each is determined 8 Abstract and Applied Analysis by the parameters and delay in (5). Thus, we can compute the following values: which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value ; that is, 2 determines the directions of the Hopf bifurcation: if 2 > 0 (< 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcation exists for > 0 (< 0 ); 2 determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable (unstable) if 2 < 0 (> 0); and 2 determines the period of the bifurcating periodic solutions: the period increases (decrease) if 2 > 0 (< 0).

Numerical Simulations
In this section, we present numerical results to verify the analytical predictions obtained in the previous sections and use the delayed-feedback controller to control the Hopf bifurcation and chaos of system (3).

Conclusion
In this paper, we have studied a delayed Lotka-Volterra predator-prey system with time delayed feedback by using the theory of functional differential equation and Hassard's method. By analyzing the corresponding characteristic equations, the local stability of the positive equilibrium of system (3) was discussed.
We have obtained the estimated length of gestation delay which would not affect the stable coexistence of both prey and predator species at their equilibrium values. The existence of Hopf bifurcation for system (3) at the positive equilibrium was also established. From theoretical analysis it was shown that the larger values of gestation time delay cause fluctuation in individual population density and hence the system becomes unstable.
As the estimated length of delay to preserve stability and the critical length of time delay for Hopf-bifurcation are dependent upon the system parameters, it is possible to impose some control, which will prevent the possible abnormal oscillation in population density. Our results show that if we choose some appropriate parameters, the oscillation can be controlled to a stable equilibrium or a stable periodic orbit; that is to say, we can achieve the ecological equilibrium by adjusting the capture (or release) level.